Quantifiers are used to express statements about all elements in a set or about the existence of at least one element. They make it possible to move from statements about individual cases to general statements, which is a central part of mathematics.

 

 

Universal quantifier (for all)

The universal quantifier indicates that something holds for all elements in a given set. It is written with the symbol \( \large \forall \).

 

Examples:

All natural numbers are greater than or equal to 0:

 

$$ \large \forall n \in \mathbb{N} : n \geq 0 $$

 

The square of a real number is not negative:

 

$$ \large \forall x \in \mathbb{R} : x^2 \geq 0 $$

 

 

Existential quantifier (there exists)

The existential quantifier indicates that there exists at least one element in the set that satisfies a certain property. It is written with the symbol \( \large \exists \).

 

Examples:

There exists a natural number which is prime:

 

$$ \large \exists n \in \mathbb{N} : n \text{ is prime} $$

 

There exists a real number whose square is 2:

 

$$ \large \exists x \in \mathbb{R} : x^2 = 2 $$

 

 

Typical misunderstandings

It is important to distinguish between the universal and existential quantifiers:

 

• \( \large \forall x \in \mathbb{N} : x \text{ is even} \) is false, because not all natural numbers are even.
• \( \large \exists x \in \mathbb{N} : x \text{ is even} \) is true, because there exists at least one natural number that is even (in fact infinitely many).

 

When multiple quantifiers appear together, the order matters greatly:

 

• \( \large \forall x \in \mathbb{R} \, \exists y \in \mathbb{R} : y = x+1 \) is true (for each number we can find another that is one greater).
• \( \large \exists y \in \mathbb{R} \, \forall x \in \mathbb{R} : y = x+1 \) is false (there is no single number that is one greater than all others).

 

 

Negation of quantifiers

Quantifiers are closely connected with negation. Negating a quantified statement means switching the quantifier and negating the inner statement:

 

$$ \large \lnot (\forall x : P(x)) \; \equiv \; \exists x : \lnot P(x) $$

$$ \large \lnot (\exists x : P(x)) \; \equiv \; \forall x : \lnot P(x) $$

 

Example: "Not all natural numbers are even" can be written as:

 

$$ \large \lnot (\forall n \in \mathbb{N} : n \text{ is even}) $$

 

This is the same as saying:

 

$$ \large \exists n \in \mathbb{N} : n \text{ is odd} $$

 

 

Summary

Quantifiers make it possible to formulate mathematical statements in general:

 

• Universal quantifier \( \forall \): something holds for all elements.
• Existential quantifier \( \exists \): there exists at least one element for which something holds.
• Under negation, quantifiers switch: "not all" becomes "there exists one that does not", and "there does not exist one" becomes "for all it does not hold".

 

These symbols are central in modern mathematics and play a major role in definitions, theorems, and proofs.